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Full text of "Analytical Mechanics"

MOTION                                      89
3.   A belt passes over a pulley which has a diameter of 30 inches and which makes 200 revolutions per minute.   Find the linear speed of the belt and the angular speed of the pulley.
4.   The wheels of a bicycle, which are 75 cm. in diameter, make 5000 revolutions in 65 minutes.    Find the speed of the rider;   the angular speed of the wheels about their axles; the relative velocity of the highest point of each wheel with respect to the center.
6. A point moves with a constant velocity v. Find its angular velocity about a fixed point whose distance from the path is a.
6.   A railroad runs due west in latitude X.   Find the velocity of the train if it always keeps the sun directly south of it.
7.   Find the expression for the angular velocity of any point on the rim of a wheel of radius a, moving with a velocity v; the wheel is supposed to be rolling without slipping.   Discuss the values of the velocity for special points.
8.   In the preceding problem find the relative velocity of any point on* the rim with respect to the center of the wheel, and the velocity of the: center with respect to the point of contact with the ground.
9.   The end of a vector describes a circle at a constant rate.    If the^ origin is outside the circle find the velocity along and at right angles to-the vector.   Discuss the values for interesting special positions.
10.   In the preceding problem derive an expression for the angular velocity of the vector and discuss it.
ACCELERATION.
86. Acceleration.—When the velocity of a particle changes it is said to have an acceleration. The change may be in the magnitude of the velocity, in the direction, or in both; further it may be positive or negative. Therefore the term acceleration includes retardation as well as increase in velocity. Retardation is negative acceleration.
If the particle moves in a straight path with a velocity which increases or diminishes at a constant rate its acceleration equals, numerically, the change in the velocity per second and is said to be constant:
f       V2~ Vl faa__,
where f is the acceleration and YI and V2 are the velocities at the beginning and at the end of the interval of time L.